Open Access

Slowly Oscillating Solutions of a Parabolic Inverse Problem: Boundary Value Problems

Boundary Value Problems20102010:471491

Received: 11 October 2010

Accepted: 20 December 2010

Published: 29 December 2010


The existence and uniqueness of a slowly oscillating solution to parabolic inverse problems for a type of boundary value problem are established. Stability of the solution is discussed.


Differential EquationContinuous FunctionIntegral EquationPartial Differential EquationUnique Solution

1. Introduction

It is well known that the space of almost periodic functions and some of its generalizations have many applications (e.g., [113] and references therein). However, little has been done for to inverse problems except for our work in [1416]. Sarason in [17] studied the space of slowly oscillating functions. This is a -subalgebra of , the space of bounded, continuous, complex-valued functions on with the supremum norm . Compared with , is a quite large space (see [1720]). What we are interested in is based on the belief that certainly has a variety of applications in many mathematical areas too. In [15], we studied slowly oscillating solutions of a parabolic inverse problem for Cauchy problems. In this paper, we devote such solutions for a type of boundary value problem.

Set . Let (resp., , where ) denote the -algebra of bounded continuous complex-valued functions on (resp., ) with the supremum norm. For (resp., ) and , the translate of by is the function (resp., , ).

Definition 1.1.
  1. (1)

    A function is called slowly oscillating if for every , , the space of the functions vanishing at infinity. Denote by the set of all such functions.

  2. (2)

    A function is said to be slowly oscillating in and uniform on compact subsets of if for each and is uniformly continuous on for any compact subset . Denote by the set of all such functions. For convenience, such functions are also called uniformly slowly oscillating functions.

  3. (3)

    Let be a Banach space, and let be the space of bounded continuous functions from to . If we replace in (1) by , then we get the definition of .


As in [17], we always assume that is uniformly continuous.

The following two propositions come from [15, Section 1].

Proposition 1.2.

Let be such that is uniformly continuous on . Then .

For , suppose that for all . Define by

The following proposition shows that the composite is also slowly oscillating.

Proposition 1.3.

Let . If and for all , then .

In the sequel, we will use the notations: , . means that is slowly oscillating in and uniformly for ; means that is slowly oscillating in and uniformly on .


be the fundamental solution of the heat equation [21].

2. A Type of Boundary Value Problem

We will keep the notation in Section 1 and at the same time introduce the following new notation:

In this section, we always assume the following: , , , , , , , , and , .


be Green's function for the boundary value problems [22, 23].

The following estimates are easily obtained:

where ( ) are positive and increasing for and as .

To show the main results of this section, the following lemmas are needed. The first lemma is Lemma 3.1 on page 15 in [24].

Lemma 2.1.

Let , , and be real, continuous functions on with . If

Lemma 2.2.

Let be a continuous function on . If , , and are nondecreasing and nonnegative on and


Replacing in the two integrals of (2.6) by the expression on the right hand side in (2.6), changing the integral order of the resulting inequality and making use of the monotonicity of , and , one gets

Apply Lemma 2.1 to get the conclusion.

Lemma 2.3.

Let , , and . Then the problem
has a unique solution , and is in and satisfies

where .

One sees that depends on only and is bounded near zero.


The existence and uniqueness of the solution comes from Theorem 5.3 on page 320 in [25].

As in [22, 23], the solution can be written as

By Lemma 2.1, one gets the desired inequality.

Now we show that . As in the proofs of Lemmas 2.1 and 2.3 in [15], one gets . For , with ,
Note that
where is a constant and
By Lemma 2.1, one has

where is a constant. Since and are slowly oscillating, the right-hand sides of the inequality above approaches zero as . This means that . The proof is complete.

Consider the following problem.

Problem 1.

Find functions and such that
One sees that
It follows from (2.24) that

Let , and let . We have the following two additional problems for and , respectively.

Problem 2.

Find functions and such that

Problem 3.

Find functions and such that

Lemma 2.4.

Problems 1, 2, and 3 are equivalent to each other.


The existence and uniqueness of the solution of Problem 2 can be easily obtained from that of the solution of Problem 1. Conversely, let ( ) be the solution of Problem 2. We show that Problem 1 has a unique solution ( ). The uniqueness comes from the uniqueness of (2.19)–(2.21). For the existence, let
Obviously, and satisfies (2.22). Also satisfies (2.21) because . By (2.23) and (2.27), one sees that (2.20) is true. Finally, we show that satisfies (2.19) and therefore, along with , constitutes a solution of Problem 1. In fact,
Thus, we have shown the equivalence of Problems 1 and 2. Replacing (2.34) by the function

the equivalence of Problems 2 and 3 can be proved similarly. The proof is complete.

By Lemma 2.4, to solve Problem 1, we only need to solve Problem 3. By (2.30)–(2.32), we have the integral equation about :
Rewrite (2.33) as

where is determined by (2.37).

One can directly test that Problem 3 is equivalent to (2.37)-(2.38).

Note that for a given , Lemma 2.3 shows that (2.30)–(2.32) (or equivalently, (2.37)) have a unique solution . Thus, (2.38) does define an operator . Therefore, we only need to show that the integral (2.38) has a unique solution and . That is, has a fixed point in . Let
Set , where . If , then, by Lemma 2.3, is in , and so, by (2.38), is in with
Equation (2.37) gives the estimate
Choose such that when , one has . It follows that
Choose such that when , one has

and therefore, .

Let , . By (2.38), . Note that the function is the solution of the problem
So, by Lemma 2.3, one has

Choose such that for , . Now, set . Then is a contraction from into itself, and therefore, has a unique fixed point. Thus, we have shown.

Theorem 2.5.

Let functions , , , and be as above. Then, for small , Problem 3 has a unique solution ( ) in with and .

Let be the solutions of Problem 3 in for the functions , , , and . Set , , , and . For the stability of the solution, we have the following.

Theorem 2.6.

For , one has

where depends on , , , , , , , and .


By (2.33),
Note that the function is the solution of the problem
Using a formula similar to (2.37) and Lemma 2.2 for the function , one gets
Applying Lemma 2.2 and (2.48), one gets the desired conclusion with
and is majorant of . One can specially assume that

The proof is complete.

Corollary 2.7.

Under the conditions in Theorem 2.6, the solution of Problem 3 is unique.



The research is supported by the NSF of China (no. 11071048).

Authors’ Affiliations

Department of Mathematics, Harbin Institute of Technology, Harbin, China


  1. Agarwal RP, de Andrade B, Cuevas C: Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations. Nonlinear Analysis: Real World Applications 2010,11(5):3532-3554. 10.1016/j.nonrwa.2010.01.002MathSciNetView ArticleMATHGoogle Scholar
  2. Andres J, Bersani AM, Grande RF: Hierarchy of almost-periodic function spaces. Rendiconti di Matematica e delle sue Applicazioni 2006,26(2):121-188.MathSciNetMATHGoogle Scholar
  3. Basit B, Zhang C: New almost periodic type functions and solutions of differential equations. Canadian Journal of Mathematics 1996,48(6):1138-1153. 10.4153/CJM-1996-059-9MathSciNetView ArticleMATHGoogle Scholar
  4. Berglund JF, Junghenn HD, Milnes P: Analysis on Semigroups: Function Spaces, Compactifications, Representations,, Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wiley & Sons, New York, NY, USA; 1989:xiv+334.MATHGoogle Scholar
  5. Bourgain J: Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations. Geometric and Functional Analysis 1996,6(2):201-230. 10.1007/BF02247885MathSciNetView ArticleMATHGoogle Scholar
  6. Corduneanu C: Almost Periodic Functions, Interscience Tracts in Pure and Applied Mathematics, no. 2. 1st edition. John Wiley & Sons, New York, NY, USA; 1968:x+237.Google Scholar
  7. Corduneanu C: Almost Periodic Functions. 2nd edition. John Wiley & Sons, New York, NY, USA; 1989.MATHGoogle Scholar
  8. Diagan T: Pseudo Almost Periodic Functions in Banach spaces. Nova Science, New York, NY, USA; 2007.Google Scholar
  9. Fink AM: Almost Periodic Differential Equations, Lecture Notes in Mathematics. Volume 377. Springer, Berlin, Germany; 1974:viii+336.Google Scholar
  10. Hino Y, Naito T, Nguyen Van Minh , Shin JS: Almost Periodic Solutions of Differential Equations in Banach Spaces, Stability and Control: Theory, Methods and Applications. Volume 15. Taylor & Francis, London, UK; 2002:viii+250.MATHGoogle Scholar
  11. N'Guérékata GM: Topics in Almost Automorphy. Springer, New York, NY, USA; 2005:xii+168.MATHGoogle Scholar
  12. Shen W: Travelling waves in time almost periodic structures governed by bistable nonlinearities. I. Stability and uniqueness. Journal of Differential Equations 1999,159(1):1-54. 10.1006/jdeq.1999.3651MathSciNetView ArticleMATHGoogle Scholar
  13. Zhang C: Almost Periodic Type Functions and Ergodicity. Science Press, Beijing, China; Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003:xii+355.View ArticleMATHGoogle Scholar
  14. Zhang C, Yang F: Remotely almost periodic solutions of parabolic inverse problems. Nonlinear Analysis: Theory, Methods & Applications 2006,65(8):1613-1623. 10.1016/ ArticleMATHGoogle Scholar
  15. Yang F, Zhang C: Slowly oscillating solutions of parabolic inverse problems. Journal of Mathematical Analysis and Applications 2007,335(2):1238-1258. 10.1016/j.jmaa.2007.01.098MathSciNetView ArticleMATHGoogle Scholar
  16. Zhang C, Yang F: Pseudo almost periodic solutions to parabolic boundary value inverse problems. Science in China. Series A 2008,51(7):1203-1214. 10.1007/s11425-008-0006-2MathSciNetView ArticleMATHGoogle Scholar
  17. Sarason D: Remotely almost periodic functions. In Proceedings of the Conference on Banach Algebras and Several Complex Variables (New Haven, Conn., 1983), Providence, RI, USA, Contemp. Math.. Volume 32. American Mathematical Society; 1984:237-242.Google Scholar
  18. Zhang C: New limit power function spaces. IEEE Transactions on Automatic Control 2004,49(5):763-766. 10.1109/TAC.2004.825970View ArticleMathSciNetGoogle Scholar
  19. Zhang C, Meng C: -algebra of strong limit power functions. IEEE Transactions on Automatic Control 2006,51(5):828-831. 10.1109/TAC.2006.875016MathSciNetView ArticleGoogle Scholar
  20. Zhang C: Strong limit power functions. The Journal of Fourier Analysis and Applications 2006,12(3):291-307. 10.1007/s00041-006-6004-2MathSciNetView ArticleMATHGoogle Scholar
  21. Friedman A: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, NJ, USA; 1964:xiv+347.Google Scholar
  22. Guo B: Inverse Problem of Parabolic Partial Differential Equations. Science and Technology Press of Heilongjiang Province, Harbin, China; 1988.Google Scholar
  23. Zauderer E: Partial Differential Equations of Applied Mathematics, Pure and Applied Mathematics. John Wiley & Sons, New York, NY, USA; 1983:xiii+779.MATHGoogle Scholar
  24. Hale JK, Verduyn Lunel SM: Introduction to Functional-Differential Equations, Applied Mathematical Sciences. Volume 99. Springer, New York, NY, USA; 1993:x+447.View ArticleMATHGoogle Scholar
  25. Ladyzenskaja OA, Solonnikov VA, Ural'ceva NN: Linear and Quasi-Linear Equations of Parabolic Type. American Mathematical Society, Providence, RI, USA; 1968.Google Scholar


© F. Yang and C. Zhang. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.